Trying to solve the Poschl-Teller potential

[haimfeld]

Trying to solve the Poschl-Teller potential (quantum mechanics)

Homework Statement

I have a superpotential that give me 2 of the Poschl-Teller potentials.
The superpotential is:
W(x)=-b*cot(x)
The Poschl-Teller potentials are:
J(x,b)=b(b-1)/(sin(x))^2-b^2
K(x,b)=b(b+1)/(sin(x))^2-b^2

Homework Equations

Schrödinger equation: Hψ=Eψ

The Attempt at a Solution

I placed the potential in Schrödinger equation (neglecting the existence of constants) and received the following equations:
ψ''(x)+(2E-J)ψ(x)=0
ψ''(x)+(2E-K)ψ(x)=0
I do not know what the next step I should do
I know I need to get to Legendre polynomial but I don't know how...
Can anyone show me how to do it?

 

[Thaakisfox]

Put $u=\cos x$. Rewrite the DE in the u variable...